Computational Mechanics https://doi.org/10.1007/s00466-019-01730-2 ORIGINAL PAPER Optimization clustering technique for PieceWise Uniform Transformation Field Analysis homogenization of viscoplastic composites Gianluca Alaimo 1 · Ferdinando Auricchio 1 · Sonia Marfia 2 · Elio Sacco 3 Received: 11 January 2019 / Accepted: 19 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Aim of the present study is to propose an enhanced method for the domain decomposition (clustering) of the representative volume element (RVE) of composite materials to be used with homogenization techniques, based on the PieceWise Uniform Transformation Field Analysis (PWUTFA). With PWUTFA, both constitutive and evolutive equations for the constituents of the composite material are written in terms of averages in each cluster; moreover, it is not required to solve via FEM the nonlinear micro-mechanical problem, allowing to drastically reduce the number of internal variables. PWUTFA is founded on the idea that it is possible to divide the RVE into large subdomains (clusters) that should group together finite elements having, under any applied loading condition, the most similar values of strain. Accordingly, in this study a multi-objective optimization-based approach is proposed with the aim to simultaneously reduce both the error in the approximation of the strain fields and the number of clusters in which the domain is decomposed. Different clustering solutions, obtained through the proposed optimization approach are analyzed, and the corresponding mechanical responses are compared with the ones obtained by the finite element analysis and by uniform transformation field analysis. Keywords RVE · Domain subdivision · Composite · PWUTFA 1 Introduction Composites materials are becoming more and more popu- lar in many fields of engineering. A composite is a material obtained from two or more constituents with significantly different physical or chemical properties. A large class of composite is made of constituents that exhibit nonlinear behavior, hence requiring the use of proper nonlinear mate- rial constitutive models. Furthermore, composite materials often present internal complex microstructures, therefore, they require specific formulations to be developed in order to take into account the mechanical behavior of each component B Gianluca Alaimo gianluca.alaimo01@universitadipavia.it 1 Dipartimento di Ingegneria Civile e Architettura, Università di Pavia, Via A. Ferrata 1, Pavia, Italy 2 Dipartimento di Ingegneria, Università di Roma Tre, Via Vito Volterra 62, 00146 Rome, Italy 3 Dipartimento di Strutture per l’Ingegneria e l’Architettura, Università di Napoli “Federico II”, Via Claudio 21, 80125 Naples, Italy and its topological distribution. Thus, the interest around the modeling of composite materials has significantly increased. In order to study the mechanical response of composite material, the micro-mechanical problem has to be solved. The overall behavior of the representative volume element (RVE) can be determined by using the finite element method (FEM), allowing to obtain accurate predictions. However, the computational effort usually associated to the RVE investi- gation via FEM is very large since it can require very fine discretizations and, consequently, a significantly large num- ber of variables introduced into the analyses. To abate such a computational effort, Reduced Order Models (ROM) can be employed, since they allow to solve the micro-mechanical problem in a reasonable computing time. Among ROM techniques, an interesting and effec- tive approach is the Transformation Field Analysis (TFA), initially introduced by Dvorak [5]. According to [5], the TFA does not require to solve via FEM the nonlinear micro-mechanical problem. In fact, the TFA considers the microscopic field of inelastic strain, representing the internal variables of the problem, uniform in the inclusions, hence drastically reducing the number of internal variables. 123